Shortly after the advent of forcing in the 1960s, Easton proved that, modulo some trivial constraints concerning monotonicity and cofinality, the axioms of set theory place no restrictions on the behavior of exponentiation at regular cardinals. In a surprising turn of events, this turned out not to be the case for singular cardinals, and the last half-century has seen a procession of deep results uncovering nontrivial constraints on exponentiation at singular cardinals. One of the first of these results was Silver's theorem, which in essence states that if \(\lambda\) is a singular cardinal of uncountable cofinality and there are "many" singular cardinals \(\kappa < \lambda\) such that \(2^\kappa = \kappa^+\), then it must also be the case that \(2^\lambda = \lambda^+\). In particular, if the Singular Cardinals Hypothesis fails, then it must fail first at a singular cardinal of countable cofinality. We will discuss this seminal theorem and a number of variations thereon, and we will end by sketching a proof of a version of Silver's theorem pertaining to certain generalized cardinal characteristics.

This talk will be given in mixed mode, in person as well as via Zoom.

If you want to attend in person, please be aware of the fact that you will be required to show proof of your COVID-19 "2.5G" status (vaccinated, recovered, PCR tested) upon entry of the buildings, or during sporadic random checks in the seminar rooms. During the lectures we will also pass around an attendance sheet to facilitate contact tracing. (According to the regulations, this form will be kept for 28 days and destroyed thereafter.)